# Light: the Quantum Model

> Black body radiation, Wien's law, Planck's quantum idea and Einstein's explanation of the photoelectric effect.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/light-quantum-model/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The light that would not push
A wave model says light is like an ocean swell: a bigger wave carries more energy. So scientists expected that very bright light would knock electrons out of a metal plate, whatever its colour, and that dim light would just need a little longer to build up enough energy.

That is not what happened. Bright red light, however bright, knocked out no electrons from zinc at all. But even the faintest ultraviolet light knocked some out instantly.

In 1905, Albert Einstein explained it. Light comes in packets. Each packet's energy is set by its colour, not its brightness. One red packet is too weak to free an electron, and a million of them are no better, because an electron only ever gets hit by one at a time. This idea won him the Nobel Prize.

**In short:** The quantum model says light comes in packets of energy called photons, each with energy $E = hf$. It explains black body radiation and the photoelectric effect, where light frees electrons from a metal only if each photon has enough energy, however bright the light is.

## Black body radiation

Every object gives off EM radiation because of its temperature. A **black body** is an ideal object that absorbs all radiation falling on it, and gives off a spectrum that depends **only on its temperature**. Stars are close to black bodies.

As a black body gets hotter:

- it gives off **more** radiation at every wavelength (the curve gets taller);
- the **peak** moves to **shorter** wavelengths (red hot, then yellow, then white, then blue-white).

### Wien's law

$$\lambda_{max} = \frac{b}{T}$$

- $\lambda_{max}$ = wavelength of peak intensity (m)
- $T$ = temperature (kelvin)
- $b = 2.898 \times 10^{-3}$ m K (Wien's constant)

### Worked example: The Sun's temperature
The Sun's spectrum peaks at about 500 nm.

$T = \dfrac{2.898 \times 10^{-3}}{500 \times 10^{-9}} = 5800$ K.

## The ultraviolet catastrophe

Classical wave theory predicted that a black body should give off more and more energy at shorter and shorter wavelengths, without limit. The curve would shoot up to infinity in the ultraviolet. This "**ultraviolet catastrophe**" did not match experiments, where the curve has a peak and falls back to zero.

## Planck's quantum idea

In 1900, Max Planck fixed the curve by assuming that energy is given off in small packets called **quanta**:

$$E = hf$$

- $h = 6.626 \times 10^{-34}$ J s (Planck's constant)
- $f$ = frequency (Hz)

High-frequency quanta need a lot of energy, so they are rare. That makes the curve fall back down at short wavelengths, just as experiments showed. Planck himself thought this was a maths trick. Einstein showed it was real.

## Photons

Einstein said light itself travels as packets of energy called **photons**. Each photon has energy:

$$E = hf = \frac{hc}{\lambda}$$

- Blue light has a higher frequency than red light, so each blue photon carries **more energy**.
- A brighter beam of the same colour has **more photons**, not more energy per photon.

### Worked example: A green photon
Green light has $\lambda = 530$ nm.

$E = \dfrac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{530 \times 10^{-9}} = 3.75 \times 10^{-19}$ J, or $\dfrac{3.75 \times 10^{-19}}{1.602 \times 10^{-19}} = 2.3$ eV.

## The photoelectric effect

When light of high enough frequency shines on a metal, electrons are knocked out. These are **photoelectrons**.

| Observation | Wave model predicts | What actually happens |
|---|---|---|
| Below a **threshold frequency** $f_0$ | brighter light should eventually work | **no** electrons, however bright |
| Above $f_0$ | electrons after a delay for dim light | electrons come out **instantly** |
| Brighter light | faster electrons | **more** electrons, same maximum speed |
| Higher frequency | no effect | **faster** electrons |

## Einstein's explanation

- One photon gives **all** its energy to **one** electron.
- The electron needs a minimum energy, the **work function** $\phi$, to escape the metal.
- Whatever energy is left over becomes the electron's kinetic energy:

$$K_{max} = hf - \phi$$

The **threshold frequency** is where the photon has just enough energy: $hf_0 = \phi$.

### Worked example: Electrons from sodium
Sodium has a work function of 2.3 eV. Light of 400 nm shines on it.

Photon energy: $E = \dfrac{hc}{\lambda} = 4.97 \times 10^{-19}$ J $= 3.1$ eV.

$K_{max} = 3.1 - 2.3 = 0.8$ eV.

## Measuring the energy: stopping voltage

In a photocell, a reverse voltage can be applied to push back on the photoelectrons. The **stopping voltage** $V_0$ is the smallest reverse voltage that stops even the fastest electrons, so the current drops to zero. Then:

$$K_{max} = qV_0$$

A stopping voltage of 0.8 V means $K_{max} = 0.8$ eV.

**Mind map**

```text
Quantum Model of Light
  Black bodies
    Spectrum depends on T
    Hotter: taller, peak shifts left
    Wien: λ max = b / T
  UV catastrophe
    Classical theory fails
  Planck
    Energy in quanta
    E = hf
  Photons
    E = hf = hc / λ
    Brighter: more photons
  Photoelectric effect
    Threshold frequency
    Instant emission
    K max = hf − φ
    Stopping voltage
```

### Year 11: Electron volts
Photon and electron energies are tiny in joules, so physicists use **electron volts**: $1 \text{ eV} = 1.602 \times 10^{-19}$ J. To change joules to eV, divide by $1.602 \times 10^{-19}$.

A photon of visible light carries about 1.8 eV (red) to 3.1 eV (violet). Metal work functions are typically 2 to 6 eV, which is why visible light only frees electrons from a few metals, such as sodium and potassium.

### Year 12: The graph of maximum kinetic energy against frequency
$K_{max} = hf - \phi$ has the form $y = mx + c$:

- The **gradient** is Planck's constant $h$, the same for **every** metal.
- The **x-intercept** is the threshold frequency $f_0$.
- The **y-intercept** is $-\phi$.

Lines for different metals are **parallel**. A metal with a bigger work function has its line shifted to the right.

On a graph of photocurrent against voltage: brighter light gives a higher current plateau but the **same** stopping voltage. Higher-frequency light gives a **bigger** stopping voltage.

**Why this changed the model of light.** Black body radiation and the photoelectric effect could not be explained by waves. Interference and diffraction could not be explained by particles. Light shows **both** behaviours: this is **wave-particle duality**.

**From energy to speed.** $K_{max} = \frac{1}{2}mv^2$, so $v = \sqrt{\dfrac{2K_{max}}{m}}$. A stopping voltage $V_0$ gives $K_{max} = qV_0$, so 1.5 V means $v = \sqrt{\dfrac{2 \times 1.602 \times 10^{-19} \times 1.5}{9.109 \times 10^{-31}}} \approx 7.3 \times 10^5$ m/s.

**Photons per second.** Divide the power by the energy of one photon: a 5.0 mW laser at 650 nm gives $\dfrac{5.0 \times 10^{-3}}{3.06 \times 10^{-19}} \approx 1.6 \times 10^{16}$ photons per second. At the same power, an infrared beam has **more** photons than an ultraviolet beam, because each infrared photon carries less energy.

**From graphs.** The threshold frequency is where the line crosses the frequency axis: $\phi = hf_0$. A graph of stopping voltage against frequency has gradient $\dfrac{h}{q}$ and crosses the frequency axis at $f_0$. Similarly, an LED glows when $qV = \dfrac{hc}{\lambda}$, so a graph of $V$ against $\dfrac{1}{\lambda}$ has gradient $\dfrac{hc}{q}$, which gives $h$.

**Halving the wavelength** doubles the photon energy, so $K_{\text{new}} = 2(K + \phi) - \phi = 2K + \phi$, more than double.

**A reverse voltage.** If photoelectrons leave with up to 2.5 eV and a 1.5 V reverse voltage slows them, they reach the anode with at most 1.0 eV.

**Hotter black bodies.** Heating from about 3000 K to 3700 K makes the curve taller at every wavelength, with its peak at a shorter wavelength.

**Using the effect.** A photoelectric sensor only works if each photon has more energy than the work function of the sensor material. A negatively charged zinc plate loses its extra electrons under ultraviolet (photon energy above zinc's work function) but not under red light of any brightness.

## Quick summary

- A black body's spectrum depends only on its temperature. Wien's law: $\lambda_{max} = \dfrac{b}{T}$.
- Classical theory failed at short wavelengths (the ultraviolet catastrophe). Planck fixed it with energy quanta $E = hf$.
- Einstein: light is made of photons, each with energy $E = hf = \dfrac{hc}{\lambda}$.
- Photoelectric effect: $K_{max} = hf - \phi$. Below the threshold frequency, no electrons are freed however bright the light.
- Brighter light means more photons and more electrons, not faster electrons.
- Stopping voltage: $K_{max} = qV_0$.

## Quiz

Try these questions from the chapter quiz.

1. As a black body gets hotter, its peak wavelength:
   - A. Gets longer
   - B. Gets shorter
   - C. Stays the same
   - D. Disappears
   - Answer: B. Gets shorter. Wien's law: λmax = b/T. Higher T gives a shorter peak wavelength.

2. Which scientist first suggested that energy is given off in small packets?
   - A. Planck
   - B. Newton
   - C. Maxwell
   - D. Young
   - Answer: A. Planck. Planck introduced quanta, E = hf, to explain black body radiation.

3. The energy of a photon depends on its:
   - A. Brightness
   - B. Frequency
   - C. Speed
   - D. Distance travelled
   - Answer: B. Frequency. E = hf. Higher frequency means more energy per photon.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/light-quantum-model/#quiz